Product formula conjecture for higher arithmetic and dynamical degrees

Let $X$ be a smooth projective variety of dimension $d$, and let $f\colon X\dashrightarrow X$ be a dominant rational self-map. For each integer $1\leqslant k\leqslant d$, let $\alpha_k(f)$ be the $k$-th arithmetic degree and let $\lambda_k(f)$ be the $k$-th dynamical degree. Product formula conjecture. One has

$$\alpha_k(f)=\max\{\lambda_k(f),\lambda_{k-1}(f)\}.$$

The paper proves the corresponding inequality in the relevant theorem and conjectures that it is always an equality; this is the proposed higher-dimensional analogue of the Kawaguchi–Silverman relation.